By [1]} there exists a minimizer \((A_m,u_m)\in \mathcal {C}_m\)  for every \(m\in \mathbb {N}\) . Without loss of generality we assume that \(\partial A_m\)  does not contain isolated points. In fact, if \(\partial A_m\)  has a isolated point \(x\)  in \(A_m^{(0)},\)  then \(A_m\setminus \lbrace x\rbrace \in \mathcal {A}_m\)  and \(\mathcal {F}(A_m,u_m) = \mathcal {F}(A_m\setminus \lbrace x\rbrace ,u_m).\)  Analogously, if \(\partial A_m\)  has an isolated point in \(A_m^{(1)},\)  then there exists \(r>0\)  such that \(B_r(x)\cap \partial A_m=\lbrace x\rbrace \)  (and \(B_r(x)\subset A_m\cup \lbrace x\rbrace  \in \mathcal {C}_m\) ). In view of Lemma REF  the function \(u_m,\)  arbitrarily extended to \(x\)  belongs to \(H_\mathrm {loc}^1(B_r(x)),\)  hence, the configuration \((A_m\cup \lbrace x\rbrace ,u_m)\in \mathcal {C}_m\)  and satisfies \(\mathcal {F}(A_m,u_m)=\mathcal {F}(A_m\cup \lbrace x\rbrace ,u_m).\)
In view of Remark REF  \((A_m,u_m-u_0)\)  is a \((\lambda _0,m)\) -minimizer of \(\mathcal {F}(\cdot ,\cdot ;\Omega ).\)  Moreover, since \(\partial A_m\)  does not contain isolated points \(\theta _*(\partial A_m,x)>0\)  for any \(x\in \partial A_m,\)  hence by
Theorem REF  the density estimates (REF ) and (REF ) hold for all \(x\in \Omega \cap \partial A_m.\)
By [1]}, there exist \(A\subset \Omega \)  and a
subsequence \(\lbrace (A_{m_h},u_{m_h})\rbrace \)  such that
\(\mathrm {sdist}(\cdot ,\partial A_{m_h})\rightarrow \mathrm {sdist}(\cdot ,\partial A)\)  as \(h\rightarrow \infty .\) 
Consider the sequence \(\mu _h:=\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits \partial A_{m_h}\)  of positive Radon measures. By Theorem  REF
\(\frac{\varsigma _*}{2} \le \frac{\mu _h( Q_r(x))}{2r} \le \frac{2\pi c_2}{c_1}\) 
for every \(x\in \Omega \cap \partial A_{m_h}\)  and \( Q_r(x)\subset \subset \Omega \)  with \(r\in (0,R_*).\)  By (REF ), (REF ) and (REF ),
\(\begin{aligned}\mu _h(\mathbb {R}^2) = \mathcal {H}^1(\partial A_{m_h}) \le &\mathcal {H}^1(\partial \Omega )+ \frac{\mathcal {F}(A_{m_h},u_{m_h}) + 2c_2\mathcal {H}^1(\Sigma )}{c_1} \\\le &\mathcal {H}^1(\partial \Omega )+ \frac{\mathcal {F}(A_1,u_1) + 2c_2\mathcal {H}^1(\Sigma ) }{c_1},\end{aligned}\) 
hence, by compactness, there exist a not relabelled subsequence and a positive Radon measure \(\mu \)  in \(\mathbb {R}^2\)  such that
\(\mu _h\rightharpoonup ^*\mu \)  as \(h\rightarrow \infty .\) 
We claim that
\(\overline{\Omega \cap \partial A} \subseteq \mathrm {supp}\,\mu \subseteq \partial A.\) 
Indeed, let \(x\in \Omega \cap \partial A\)  and \(r\in (0,\min \lbrace \mathrm {dist}(x,\partial \Omega ),R_*\rbrace ).\) 
By the \(\mathrm {sdist}\) -convergence, there exists \(x_h\in Q_r(x)\cap \partial A_{m_h}\)  with \(x_h\rightarrow x\)  such that \( Q_{r/2}(x_h)\subset Q_r(x)\)  and hence, by the weak* convergence and (REF ),
\(\mu (\overline{ Q_r(x)}) \ge \limsup \limits _{h\rightarrow \infty }\mu _h(\overline{ Q_r(x)})\ge \limsup \limits _{h\rightarrow \infty }\mu _h( Q_{r/2}(x_h)) \ge \varsigma _* r.\) 
This implies \(x\in \mathrm {supp}\,\mu .\) 
Conversely,
if, by contradiction, there exists \(x\in \mathrm {supp}\,\mu \setminus \partial A,\) 
then we can find \(r>0\)  such that \( Q_r(x)\cap \partial A=\emptyset .\)  From the \(\mathrm {sdist}\) -convergence
it follows that \( Q_{r/2}(x)\cap \partial A_{m_h}=\emptyset \)  for \(h\)  large enough, and hence,
\(0<\mu ( Q_{r/2}(x)) \le \liminf \limits _{h\rightarrow \infty } \mu _h( Q_{r/2}(x)) =0,\) 
which is a contradiction.
From (REF ) it follows that
\(\frac{\varsigma _*}{2} \le \frac{\mu ( Q_r(x))}{2r} \le \frac{2\pi c_2}{c_1}\) 
for any \(x\in \Omega \cap \mathrm {supp}\,\mu \)  any \(r\in (0,R_*)\)  with \( Q_r(x)\subset \subset \Omega \) . Indeed, let \(x\in \Omega \cap \mathrm {supp}\,\mu \)  and let \(R(x):=\min \lbrace R_*,\mathrm {dist}(x,\partial \Omega ).\)  Then by the weak* convergence \(\mu _h(Q_r(x))\rightarrow \mu (Q_r(x))=0\)  for a.e. \(r\in (0,R(x)).\)  In particular, (REF ) holds for a.e. \(r\in (0,R(x)).\)  Since \(\mu \)  is a Radon measure, (REF ) extends to all \(r\in (0,R(x))\)  by the left-continuity of the map \(r\mapsto \mu (Q_r(x))\) .
From (REF ) and [3]} it follows that
\(\varsigma _*\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\Omega \cap \mathrm {supp}\,\mu ) \le \mu \mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits \Omega \le \frac{4\pi c_2}{c_1}\,\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\Omega \cap \mathrm {supp}\,\mu ).\) 
Thus, \(\color {black}\mu \mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits \Omega \)  is absolutely continuous with respect to \(\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\Omega \cap \mathrm {supp}\,\mu )\)  and \(\mathcal {H}^1(\mathrm {supp}\,\mu )<\infty .\) 
By (REF ),
\(\mathcal {H}^1(\partial A) \le \mathcal {H}^1(\Omega \cap \partial A) +\mathcal {H}^1(\partial \Omega \cap \partial A) \le \frac{1}{\varsigma _*}\,\mu (\Omega ) +\mathcal {H}^1(\partial \Omega ) <\infty .\) 
In view of (REF ) for any \(x\in \Omega \cap \partial A\)
\(\varsigma _* <\limsup \limits _{r\rightarrow 0} \frac{\mu (Q_r(x))}{r} \le \frac{2\pi c_2}{c_1\varsigma _*}\, \liminf \limits _{r\rightarrow 0} \frac{\mu (Q_r(x))}{r} <\frac{4\pi ^2c_2^2}{c_1^2\varsigma _*}.\) 
Therefore by
Finally let us prove (REF ). Fix any \(x\in \Omega \cap \partial A\)  and let \(R(x):=\min \lbrace R_*,\mathrm {dist}(x,\partial \Omega )\rbrace .\)  Then by (REF )
\(\frac{\varsigma _* \mathcal {H}^1(Q_r(x)\cap \partial A)}{2r} \le \frac{\mu (Q_r(x))}{2r} \le \frac{4\pi c_2}{c_1}\,\frac{\mathcal {H}^1(Q_r(x)\cap \partial A)}{2r}.\) 
This and (REF ) imply
\(\frac{\varsigma _* \mathcal {H}^1(Q_r(x)\cap \partial A)}{2r} \le \frac{2\pi c_2}{c_1}\quad \text{and}\quad \frac{\varsigma _*}{2}\le \frac{4\pi c_2}{c_1}\,\frac{\mathcal {H}^1(Q_r(x)\cap \partial A)}{2r},\) 
and hence, (REF ) follows.
